Fourier series is just a means to represent a periodic signal as an infinite sum of sine wave components. A periodic signal is just a signal that repeats its pattern at some period. The primary reason that we use Fourier series is that we can better analyze a signal in another domain rather in the original domain. Sometimes a signal reveals itself more in another domain. Any periodic signal is just a combination of sine and cosine waves. Fourier series is a method to represent that periodic signal that we can manage it easily with simple trigonometry skills.
If any periodic function can be expressed as the sum of a finite or infinite number of sinusoidal functions, the responses of linear networks to non-sinusoidal excitation can be determined by applying superposition theorem.
What do you mean by periodic function?
A periodic function is that which repeat itself after regular intervals of time.
A function f(t) is said to be periodic if f(t) = f(t+T) for every t, where T is called the period of the function.
If n as an integer if follows that
f(t) = f(t+nT) =f(t - nT)
Evaluation of Fourier Constants:
In general, a periodic signal can be represented as a sum of both sines and cosines. Also, since sines and cosines have no average term, periodic signals that have a non-zero average can have a constant component. Altogether, the series becomes the one shown below. This series can be used to represent many periodic functions. The function, f(t), is assumed to be periodic.
The coefficients, an and bn, are what you need to know to generate the signal.
To compute the coefficients we take advantage of some properties of sinusoidal signals. The starting point is to integrate a product of f(t) with one of the sinusoidal components as shown below.
Now, if we assume that the function, f(t), can be represented by the series above, we can replace f(t) with the series in the integral.
Here, we note the following:
f = 1/T,
wo = 2pf.
Then, when we do the integration, we can use some properties of the sinusoidal functions. In all cases here, the integral is take over exactly one period of the periodic signal, f(t).
So, when we do the integration of the function, f(t), multiplied by any sine or cosine function, they almost all work out to be zero. The only one that doesn't work out to be zero is the one where n and m are equal.
Realizing all of this, we can conclude:
Which gives us a way to compute any of the b's in the Fourier Series. At this point we have half of our problem solved because we can compute the b's, but we still need to compute the a's. However, we can do the same thing for the a's that we did for the b's (and we will let you do that yourself) and we get the following expressions for the coefficients.
and these expressions are good for n>0 and m> 0. The only coefficient not covered is ao which is given by: So, now we have a way to find all of the coefficients in a Fourier Series expansion.
Waveform Symmetry:
For some waveforms it is not necessary that Fourier Series contain all sine and cosine terms. This is because of symmetry exhibited by the waveform. Knowledge of such symmetry results in reduced calculations in determining Fourier Series. Following presentation describes various waveform symmetry and simplified value of calculating coefficients.
Introduction:
Fourier series is just a means to represent a periodic signal as an infinite sum of sine wave components. A periodic signal is just a signal that repeats its pattern at some period. The primary reason that we use Fourier series is that we can better analyze a signal in another domain rather in the original domain. Sometimes a signal reveals itself more in another domain. Any periodic signal is just a combination of sine and cosine waves. Fourier series is a method to represent that periodic signal that we can manage it easily with simple trigonometry skills.
If any periodic function can be expressed as the sum of a finite or infinite number of sinusoidal functions, the responses of linear networks to non-sinusoidal excitation can be determined by applying superposition theorem.
What do you mean by periodic function?
A periodic function is that which repeat itself after regular intervals of time.
A function f(t) is said to be periodic if f(t) = f(t+T) for every t, where T is called the period of the function.
If n as an integer if follows that
f(t) = f(t+nT) =f(t - nT)
Evaluation of Fourier Constants:
In general, a periodic signal can be represented as a sum of both sines and cosines. Also, since sines and cosines have no average term, periodic signals that have a non-zero average can have a constant component. Altogether, the series becomes the one shown below. This series can be used to represent many periodic functions. The function, f(t), is assumed to be periodic.
The coefficients, an and bn, are what you need to know to generate the signal.
To compute the coefficients we take advantage of some properties of sinusoidal signals. The starting point is to integrate a product of f(t) with one of the sinusoidal components as shown below.
Now, if we assume that the function, f(t), can be represented by the series above, we can replace f(t) with the series in the integral.
Here, we note the following:
- f = 1/T,
- wo = 2pf.
Then, when we do the integration, we can use some properties of the sinusoidal functions. In all cases here, the integral is take over exactly one period of the periodic signal, f(t).So, when we do the integration of the function, f(t), multiplied by any sine or cosine function, they almost all work out to be zero. The only one that doesn't work out to be zero is the one where n and m are equal.
Realizing all of this, we can conclude:
Which gives us a way to compute any of the b's in the Fourier Series.
At this point we have half of our problem solved because we can compute the b's, but we still need to compute the a's. However, we can do the same thing for the a's that we did for the b's (and we will let you do that yourself) and we get the following expressions for the coefficients.
and these expressions are good for n>0 and m> 0. The only coefficient not covered is ao which is given by:
So, now we have a way to find all of the coefficients in a Fourier Series expansion.
Waveform Symmetry:
For some waveforms it is not necessary that Fourier Series contain all sine and cosine terms. This is because of symmetry exhibited by the waveform. Knowledge of such symmetry results in reduced calculations in determining Fourier Series. Following presentation describes various waveform symmetry and simplified value of calculating coefficients.